Lightly butter the bottom and sides of a 9×13 inch baking pan and set it aside.

In a small pot, melt 4 1/2 tablespoons of unsalted butter with 2 tablespoons of minced garlic over low heat for 1 to 2 minutes until fragrant, then remove from heat to cool.

In a stand mixer bowl, combine 3 1/2 cups plus 1 tablespoon flour, 3 tablespoons parsley, 1 tablespoon sugar, 2 1/4 teaspoons yeast, and 1 teaspoon salt.

Add warmed milk, 1 large egg, and cooled garlic butter mixture to the dry ingredients.

Knead the dough with a dough hook on low speed for about 10 to 12 minutes until smooth and slightly tacky, adding more flour as needed.

Divide the dough into 12 equal portions and shape each into a smooth ball.

Arrange the dough balls in the prepared pan, cover with plastic wrap, and let rise in a warm spot for 1 to 2 hours until doubled in size.

Preheat your oven to 350°F (180°C).

Brush the tops of the risen rolls with the beaten egg for a glossy finish.

Bake rolls for 20 to 30 minutes until golden brown and sound hollow when tapped.

Meanwhile, melt 2 tablespoons butter and stir in 1 tablespoon minced garlic, then remove from heat and mix in 1 tablespoon parsley.

Brush the baked rolls with the garlic butter mixture and sprinkle with flaky sea salt.